The embedding theorem for finite depth subfactor planar algebras

We define a canonical relative commutant planar algebra from a stronglyMarkov inclusion of finite von Neumann algebras. In the case of a connectedunital inclusion of finite dimensional C*-algebras with the Markov trace, weshow this planar algebra is isomorphic to the bipartite graph planar algebra ofthe Bratteli diagram of the inclusion. Finally, we show that a finite depthsubfactor planar algebra is a planar subalgebra of the bipartite graph planaralgebra of its principal graph.